Mathematics and Philosophy

Date: 05/15/99 at 13:52:43
From: Wim Schreurs
Subject: Mathematics and truth

Excuse my English, please.

1. I think mathematics made a mistake. 

2. I think that mathematics cannot give an exact representation of 
reality in the sense of perceptibility - even things that I think 
myself are perceptible because I can 'see' or 'hear' the things I 
think, because if I don't, I don't think.

3. So there is no reality beyond perceptibility. There is only 
perceptibility.

4. You cannot say 2 + 3 = 5, because 2 is not three is not five. How 
can two things (2 and 3), neither of which is identical to 5, be 
identical to five if they are united? 5 in itself is also 
an independent "being." If it weren't, it indeed could exist as a 
collection of at least two other 'entities'.

5. Conclusion: only 5 = 5 or 2 = 2 is a correct mathematical 
exercise. ALL the other mathematic "exercises" are wrong. Something 
can only be identical to itself.

6. Question (although I know the answer): Is this true or not?

7. Where do they examine this terrible problem further? I'm afraid of 
this theory. Help me please.

Wim Schreurs

Date: 05/17/99 at 16:55:04
From: Doctor Peterson
Subject: Re: Mathematics and truth

Hello, Wim. You have some deep thoughts here, and I'll try to help you 
think through them.

Mathematics deals not with reality, but with an abstraction of 
reality: a "model" of just one aspect of the reality we use it to 
describe. For example, a number such as 2 doesn't represent any 
particular pair of things, but the idea of "two-ness." Man has found 
through long experience that things can be counted, and that the 
resulting numbers accurately describe one aspect of reality: if I 
counted two apples yesterday, and nothing has been done to them, then 
when I count them again there will still be two. The number doesn't 
tell us their color, or how they taste, or who owns them; but it 
describes something about them that is true of any pair of apples. It 
is as much something I perceive about them as is their color or taste; 
but when I talk about numbers, I am abstracting one property from the 
rest, thinking only of the "two-ness" and ignoring the "apple-ness."

Now if I have the two apples I counted yesterday, and you give me 
three more apples, I can count them again and I know there will be 
five. I don't need to know anything else about the apples to know 
this; addition is a property of the numbers themselves, independent of 
any other properties the apples may have. This means I can forget 
about the apples themselves, and even, if I wish, forget about the 
process of counting and the perception involved in doing that. Here we 
have entered the realm of mathematics, where we deal with abstract 
numbers rather than specific counted items. And within this realm, 
since we are no longer dealing with perception, what we say can be 
exactly true - although when we take our conclusions back to the 
"real" world, we will have to deal with the possibility that our 
perceptions are inaccurate: we counted wrong, apples can disappear 
spontaneously, or whatever. I might try to apply addition to something 
for which it doesn't work (adding, say, a liter of sugar to a liter of 
water, and expecting 2 liters of sugar water); then the problem is not 
in the math but in the application. It is the job of science (or 
merely of experience) to determine what mathematical models apply to a 
given situation.

Returning to the mathematical realm, when I say that 2+3 = 5, I am not 
saying that any particular 2 and 3 items are the same as any 
particular 5 items. I am saying, in an abbreviated form, that whenever 
I put together 2 of something and 3 more of them, I will have 5 in 
all. Also, I am not somehow "combining" a "2" and a "3" to make 
something new that is identical to "5"; I am performing an operation 
on two numbers that is the abstract representation of putting together 
two sets of things, in order to determine the number of things in the 
combined set. If you prefer, you can say I am simply stating a 
relation among the numbers 2, 3, and 5.

"Equal" does not mean "identical"; it means that they are two 
different ways to describe the same reality: I can view the apples as 
a set of two and a set of three, or as one set of five. If I walk 
forward two steps, wait a minute, then walk forward three steps, I 
will be in the same place as if I had walked five steps all at once. 
So "2+3" and "5" are "equal" in the sense that they have the same 
meaning, or produce the same result, even though they are different 
symbols.

You could certainly invent your own meaning for mathematical terms, 
so that "equals" could only be used between two of the same number 
(2 = 2). But that would be worthless, since it would not allow any 
calculations to be done; and it would neglect millennia of human 
experience that calculations can in fact be done. You could also say 
that numbers have no reality in themselves, and that is in a sense 
true. What makes math "work" is that numbers, which are not a part of 
the "real" world, nevertheless describe it very accurately. It doesn't 
really matter whether numbers are "real" or whether they are an "exact 
representation" of reality (they are correct, but not complete); they 
are simply a language that describes one aspect of the reality we 
perceive.

Now, if you are concerned that there may be no reality beyond your own 
perception, I think you can relax. The kind of philosophy you are 
struggling with has trouble dealing with the surprising "reality" of 
math, among other things. The fact that this mathematical "language," 
which fits together so beautifully, turns out to fit the reality we 
perceive, should encourage us to believe that what we perceive has an 
order to it far deeper than our own imagination could build (do your 
dreams have such order and consistency?), and therefore has some 
reality beyond ourselves. Most of us, because we don't think deeply, 
aren't surprised that math "works"; you have recognized that there is 
no reason to assume that the "reality" of math and the "reality" of 
perception would fit together at all, so math becomes a surprise that 
makes you re-evaluate your philosophy. That's good.

Please feel free to write back if you need more help thinking about 
this. Deep thinking like this can be scary, and it's good not to do it 
alone.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/